## Newly added calculators

.....The site is being constantly updated, so come back to check new updates.....

If you find any bug or need any improvements in solution report it here

Crypto fear and greed index

# Moment generating function calculator

This calculator is for finding MGF from PDF of a continuous random variable. The moment-generating function is useful for finding the mean and variance of the distribution.

INSTRUCTION: If you click on step function input box. A keyboard will appear. Use that keyboard to enter your function

Step functions Upper and lower limit of step function

## You can see a sample solution below. Enter your data to get the solution for your question

$$\displaylines{---}$$
$$\displaylines{ \mathbf{\color{Green}{Function\;you\;entered\;is,\;}} \mathbf{\color{Red}{F(x)}} =\left\{\begin{matrix} \mathbf{\color{Red}{0}} & -\infty <x< 0.0 \\ \\ \mathbf{\color{Red}{x}} & 0.0 ≤x≤ 1.0 \\ \\ \mathbf{\color{Red}{2 - x}} & 1.0 ≤x≤ 2.0 \\ \\ \mathbf{\color{Red}{0}} & 2.0 <x< \infty \\ \\ \end{matrix}\right. \\ \\ Moment\;generating\;function,\;M_{X}(t)\;=\; \int_{ -\infty }^{ \infty } (e^{t x} f{\left(x \right)})dx \\ \\ \Rightarrow \int_{ -\infty }^{ 0.0 } (0)dx+\int_{ 0.0 }^{ 1.0 } (x e^{t x})dx+\int_{ 1.0 }^{ 2.0 } (\left(2 - x\right) e^{t x})dx+\int_{ 2.0 }^{ \infty } (0)dx \\ \\ \Rightarrow 0+\begin{cases} \frac{\left(1.0 t - 1\right) e^{1.0 t}}{t^{2}} + \frac{1}{t^{2}} & \text{for}\: t > -\infty \wedge t < \infty \wedge t \neq 0 \\0.5 & \text{otherwise} \end{cases}+\begin{cases} - \frac{\left(1.0 t + 1\right) e^{1.0 t}}{t^{2}} + \frac{e^{2.0 t}}{t^{2}} & \text{for}\: t > -\infty \wedge t < \infty \wedge t \neq 0 \\0.5 & \text{otherwise} \end{cases}+0 \\ \\ \Rightarrow \begin{cases} \frac{\left(- 1.0 t - 1\right) e^{1.0 t}}{t^{2}} + \frac{e^{2.0 t}}{t^{2}} & \text{for}\: t > -\infty \wedge t < \infty \wedge t \neq 0 \\0.5 & \text{otherwise} \end{cases} + \begin{cases} \frac{\left(1.0 t - 1\right) e^{1.0 t}}{t^{2}} + \frac{1}{t^{2}} & \text{for}\: t > -\infty \wedge t < \infty \wedge t \neq 0 \\0.5 & \text{otherwise} \end{cases} }$$